Free line arrangements with low maximal multiplicity
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917360284205056 |
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| author | Dimca, Alexandru Kühne, Lukas Pokora, Piotr |
| author_facet | Dimca, Alexandru Kühne, Lukas Pokora, Piotr |
| contents | Let $\A$ be a free arrangement of $d$ lines in the complex projective plane, with exponents $d_1\leq d_2$. Let $m$ be the maximal multiplicity of points in $\A$. In this note, we describe first the simple cases $d_1 \leq m$. Then we study the case $d_1=m+1$, and describe which line arrangements can occur by deleting or adding a line to $\A$. When $d \leq 14$, there are only two free arrangements with $d_1=m+2$, namely one with degree $13$ and the other with degree $14$. We study their geometries in order to deepen our understanding of the structure of free line arrangements in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01733 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free line arrangements with low maximal multiplicity Dimca, Alexandru Kühne, Lukas Pokora, Piotr Algebraic Geometry Combinatorics 14H50, 32S25, 13D02 Let $\A$ be a free arrangement of $d$ lines in the complex projective plane, with exponents $d_1\leq d_2$. Let $m$ be the maximal multiplicity of points in $\A$. In this note, we describe first the simple cases $d_1 \leq m$. Then we study the case $d_1=m+1$, and describe which line arrangements can occur by deleting or adding a line to $\A$. When $d \leq 14$, there are only two free arrangements with $d_1=m+2$, namely one with degree $13$ and the other with degree $14$. We study their geometries in order to deepen our understanding of the structure of free line arrangements in general. |
| title | Free line arrangements with low maximal multiplicity |
| topic | Algebraic Geometry Combinatorics 14H50, 32S25, 13D02 |
| url | https://arxiv.org/abs/2505.01733 |